Department of Electrical Engineering and Information Technology. Institute for Communications and Navigation Prof. Dr. sc. nat.

Transcription

1 Technische Universität München Department of Electrical Engineering and Information Technology Institute for Communications and Navigation Prof. Dr. sc. nat. Christoph Günther Master thesis Partial ambiguity fixing for precise point positioning with multiple frequencies in the presence of biases Supervisor: Author: Started: Completed: Dipl.-Ing. Patrick Henkel Víctor Danubio Gómez Pantoja

2 Contents Acknowledgement iii 1 Introduction 1 2 Global Navigation Satellite Systems GNSS projects GNSS Measurements Code measurements Carrier phase measurements Measurement s errors and delays Carrier Phase Positioning and Integer Ambiguity Resolution Simplification methods for ambiguity resolution Linear combinations GP-IF-NP Four-frequency Galileo code-carrier combination Carrier smoothing Satellite-satellite single-difference (SD) Linear model for position estimation Integer estimation Integer estimators Quality of ambiguity estimates Success rate of the bootstrapped estimator Success rate bounds Integer ambiguity resolution in the presence of biases Biased-affected bootstrapped success rate Absolute carrier phase positioning model Sequential fixing i

3 CONTENTS ii 4.4 Bounding the biases Exponential bias profile Bounding of conditional biases Partial integer decorrelation Partial ambiguity fixing Partial ambiguity fixing methods PRN order method Sequential fixing Ascending Variance Order (SAVO) method Sequential BLewitt fixing Order (SEBLO) method Sequential Optimum Fixing Order Search (SOFOS) method Rounding fixing method Comparison of the partial ambiguity fixing methods SOFOS and SAVO methods SOFOS and SEBLO methods SOFOS and Rounding fixing methods Summary and conclusions 76 A Differencing 78 A.1 Single difference A.2 Double difference B Cardano s method 80 C Kronecker product 82 D Further SOFOS simulations 83 Bibliography 86

4 Acknowledgement I would like to thank Prof. Dr. sc. nat. Christoph Günther for the cordially incorporation to his team and for giving me the chance to fulfill my master thesis on this interesting research topic. I am greatly thankful to my supervisor, Patrick Henkel, for the allocation of this very interesting topic, for his always friendly and excellent provided support, and for his singular suggestions and ideas to improve this work. My sincere thanks to Kaspar Giger and Vladimir Kuryshev for their help and friendly discussions. Besides, I would like to thank the family of my girlfriend Linda, particularly her parents, for all their support in Germany. Especially, I want to thank my family, mainly my parents for all their support, trust and affection, as well as my sisters, Silvia and Perla, and my brothers, Goyo and Héctor, for all their help and advice. Finally, I want to express my special gratitude to mi lovely pequeñita Linda for her support in all respects and invaluable company. iii

5 Chapter 1 Introduction The use of Global Navigation Satellite Systems (GNSS) for navigation, surveying and geophysics has continuously increased over the last decades. However, not only the demand has increased but also the requirements on accuracy, precision, reliability, availability and continuity of the systems. Therefore, a new generation of GNSS is built, the European Galileo system, and a modernization of the first and current Global Positioning System (GPS) is also on track. In order to meet the more stringent with respect to precision, integrity and real-time positioning, carrier phase measurements are used. These measurements are extremely precise but ambiguous by an unknown number of cycles, which are known to be integer-valued. For solving this problem integer ambiguity resolution algorithms have been developed. The integer estimation process usually consists of three steps: First, a standard least-squares method is applied by disregarding the integer property of the ambiguities and the so-called float solution is obtained. In the second step, the integer constraint of the ambiguities is considered, i.e. the float ambiguities are mapped to integer values. Different choices for this mapping are available. The most simple, for example, is by rounding to the nearest integer value. However, for this choice between different estimators, the probability of correct integer estimation should be taken into account. Finally, the fixed integer-valued ambiguities are use to adjust the remaining unknown parameters by their correlation. The least-squares integer ambiguity decorrelation adjustment (LAMBDA) proposed by Teunissen in [1] is widely applied for integer ambiguity resolution. After obtaining the float solution, an integer-valued ambiguity decorrelation transformation is computed from the covariance matrix of the float ambiguities by alternating integer-approximated Gaussian eliminations and permutations of ambiguities (de Jonge and Tiberius [2]). Afterwards, a 1

6 Chapter 1. Introduction 2 search and a back-transformation from the decorrelated search space into the original ambiguity space is performed by using the inverse of the decorrelation matrix. This integer decorrelation transformation has been originally derived for unbiased measurements where it performs very well; however, it becomes critical for biased measurements as it amplifies the biases, which affects the reliability of the ambiguity resolution. Therefore, a partial integer decorrelation is suggested to achieve an optimum trade-off between the noise variance reduction and the biases amplification. The LAMBDA method was originally used for double-difference (DD) carrier phase measurements. However, it can be applied also to single-difference (SD) measurements. A successful resolution of all ambiguities, also known as Full Ambiguity Resolution, may not always be possible (i.e. the probability of correct integer estimation is too low). Severe multipath or large uncorrected biases may prevent this reliable resolution. However, a reliable resolution of a subset of the ambiguities can still be possible, also referred to as Partial Ambiguity Resolution and introduced by Teunissen et al. in [3]. The aim of partial ambiguity resolution is to identify the subset of ambiguities which gives the largest possible probability of correct integer estimation. Partial ambiguity fixing for double-differences have been analyzed by Cao et al. in [4] with a short baseline. On this thesis, different partial ambiguity fixing methods in the presence of biases are investigated for carrier phase absolute positioning with satellite-satellite single-difference measurements at a single epoch. A new partial ambiguity fixing method is used for sequential fixing, which considers all the possible orders of fixing to obtain an optimum largest subset of fixable ambiguities. This new method is the so-called Sequential optimum fixing order search (SOFOS) method. Moreover, a new exponential profile is suggested to upper-bound the residual biases. This thesis is outlined as follows. In Chapter 2, a brief introduction to the current and future Global Navigation Satellite Systems is given. The GNSS measurement model is also introduced. In Chapter 3 absolute carrier phase positioning and the ambiguity resolution problem are described. Three different methods to simplify and improve the ambiguity resolution are discussed: Satellite-satellite single differences (SD), linear combinations and carrier smoothing. The derivation of a new four-frequency code-carrier linear combination for Galileo is also presented. Furthermore, three integer estimators are briefly described: integer rounding, integer bootstrapping and integer least-squares. Finally, the quality of the ambiguity estimates is analyzed.

7 Chapter 1. Introduction 3 The presence of biases on the GNSS measurements and how they affect the ambiguity resolution is analyzed in Chapter 4. A biased carrier phase absolute positioning model using the simplification methods from the previous chapter is discussed. Moreover, an upperbound for the biases is derived by using an exponential bias profile. Furthermore, due to the amplification of the biases from the ambiguity decorrelation transformation matrix of the LAMBDA method, a partial integer decorrelation is discussed. Finally, partial ambiguity fixing is introduced for severe multipath and large uncorrected biases. In Chapter 5 different partial ambiguity fixing methods using sequential and batch fixing are described. The methods are implemented in Matlab R, and simulations are used to investigate their performance. The comparison of their performance is presented in Chapter 6. Finally, the conclusions are summarized and an outlook is given in Chapter 7.

8 Chapter 2 Global Navigation Satellite Systems Global Navigation Satellite Systems (GNSS) are built of medium earth orbit (MEO) satellites that provide a global coverage for positioning with applications in navigation, surveying, location-based services and geophysics. Four global navigation systems GPS, Galileo, GLONASS and Compass will be available within the next years. A brief description of them is given below. The frequency allocations for their signals (Misra and Enge [5], InsideGNSS [6], and Verhagen [7]) are illustrated in Figure GNSS projects GPS The Global Positioning System (GPS), developed by the U.S. Air Force in the 1970 s, is the first operational GNSS, and is currently the most utilized satellite navigation system on the world. The fully operational GPS constellation consists of 24 MEO satellites, divided over 6 orbital planes, but it can uses up to 32 satellites. The orbital inclination angle is 55, with an orbital radius of 26,600 km. GPS transmits currently on two radio frequencies in the L-band using code division multiple access CDMA techniques. On the L1 (Link 1) band, centered at MHz, two signals are transmitted, the free coarse acquisition (C/A) code for civil users, and the military precision P-code for military users. One P-code for military users is transmitted at the L2 band, centered at MHz. Civil users can only access the P-code encrypted form (Y-code). The frequency band L5, centered at MHz, is being added to the system, under the GPS modernization plan, and the codes transmitted on it will be freely available. In 4

9 Chapter 2. Global Navigation Satellite Systems 5 addition, a civil signal is being introduced at the L2 band, and a modernized M-code signal for military users is currently being implemented on L1 and L2 bands. Figure 2.1: Frequency allocations for GPS, GLONASS, Galileo and Compass. GLONASS The Russian acronym GLONASS stands for GLObal naya NAvigationsnaya Sputnikovaya Sistema, which literally is the Russian s GNSS. It is chronologically the world s second GNSS system. The satellite constellation consists of 24 satellites and can be described compactly as a Walker 24/3/2 with an orbital inclination angle of 64.8, and an orbital radius of 25,500 km. GLONASS transmits also on the L1 ( MHz) and L2 ( MHz) frequency bands using frequency division multiple access FDMA techniques. Due to the fall of the Soviet Union and the followed economical collapse, a full GLONASS constellation was available only for a short time in 1995; thereafter, it declines to only seven operational satellites in That year, a GLONASS modernization program was initiated. It is expected a fully operational constellation by Furthermore, the addition of a third frequency in the band MHz as well as a CDMA signal is being evaluated. Note: A Walker constellation (Walker T/P/F ), represents a satellite constellation of T satellites in circular orbits, divided over P equally spaced orbital planes. The satellites are evenly distributed in each orbit and the relative spacing of the satellites between adjacent planes is F in units of 360 /T.

10 Chapter 2. Global Navigation Satellite Systems 6 Galileo Galileo is the currently under development European s Union GNSS. In contrast to GPS, it will be an independent and civilian system. The project has been led by a partnership between the European Commission (EC) and the European Space Agency (ESA). It will have international participation and investment, and will be interoperable and compatible with GPS and GLONASS systems. The Galileo constellation, compactly described as a Walker 27/3/1, will consist of 30 satellites, divided over 3 orbital planes (9 operational satellites and one active spare). The orbital inclination will be 56 and the orbital radius of 29,600 km. Galileo signals will be transmitted using CDMA techniques on four frequency bands: E5a ( MHz), E5b ( MHz), E6 ( MHz) and E2-L1-E1 ( MHz). Services Galileo signals will be assigned to provide the following types of services which are summarized in Table 2.1. (Ávila et al. [8]), Open Service (OS): Free-accessible use for anyone, providing basic navigation. Commercial Service (CS): Fee-based service offering additional commercial data, service availability and higher accuracy than the OS. Safety-of-Life Service (SoL): Fee-based service aimed to safety-critical transport applications (e.g. air-traffic control) offering integrity reliability and authentication of the signal, certification and guarantee of service. Public Regulated Service (PRS): Fee-based service aimed to security authorities and military applications offering the same services as SoL. Additionally, it uses encrypted PRS codes and anti-jamming signals. Search And Rescue Service (SAR): Used for detection of distress alerts and coordination of search-rescue teams. Table 2.1: Galileo services mapped to signals. Frequency band OS CS SoL PRS SAR E5a E5b E6 E1

11 Chapter 2. Global Navigation Satellite Systems 7 Compass Compass is a GNSS project to be developed by China. The Compass constellation will consist of 35 satellites (5 geostationary orbit (GEO) satellites and 30 MEO satellites). Compass signals will be transmitted on the E1 ( MHz), E2 ( MHz), E5b ( MHz) and E6 ( MHz) frequency bands. It will offer an open and a restricted service. 2.2 GNSS Measurements Two types of measurements result from processing the GNSS signals: code and carrier phase measurements Code measurements Code tracking provides a coarse measure of the distance between the satellite and the receiver, also referred to as pseudorange. It estimates the apparent transit time of the signal, defined as the difference between the signal reception time at the receiver and the signal transmission time at the satellite. The corresponding pseudorange is defined as the given transit time multiplied by the speed of light in vacuum ρ(t) = c[t u (t) t k (t τ)], (2.1) where t u (t) is the reception time at user u; t k (t τ) is the transmission time from satellite k; c is the speed of light in vacuum; and τ is the travel time of the signal. This measurement is biased by the receiver and satellite clocks, which are not equal and also differ from the GPS Time (GPST). Taking into account these errors into equation (2.1) yields ρ(t) = c[(t + δt u (t)) (t τ + δt k (t τ))] = c[τ + (δt u (t) δt k (t τ)], (2.2) where δt u (t) and δt k (t τ) are the receiver and satellite clock offsets, respectively. Atmospheric effects (ionospheric and tropospheric delays), multipath, orbital errors and noise also affect the code measurements. A brief explanation of them will be given in Section Extending the code measurement model of user u, satellite k on frequency m and epoch i

12 Chapter 2. Global Navigation Satellite Systems 8 yields ρ k u,m(t i ) = ru(t k i ) + Tu k (t i ) + q1mi 2 u(t k i ) + δru(t k i ) +c(δτ u (t i ) δτ k (t i )) + b ρu,m + b k ρ m + ε k ρ u,m (t i ), (2.3) where ru k = cτ, is the satellite-user range; Tu k is the tropospheric delay; Iu k is the ionospheric delay depending on the ratio of frequencies q 1m ; δru k is the projected satellite orbital error; b ρu,m and b k ρ m are the receiver and the satellite code biases including instrumental delays; ε k ρ u,m is the code noise including multipath; δτ u,m is the receiver clock error; and δτ k m is the satellite clock error Carrier phase measurements The carrier phase measurement is much more precise than the code measurement. It is defined as the difference between the phase of the carrier received from the satellite and the phase of the receiver-generated carrier signal at the instant of the measurement. The phase is measured in terms of the number of cycles generated or received since the starting point at zero time. Then, the carrier phase measurement will be the measured fractional cycle plus an unknown number of whole cycles, which is also called integer ambiguity. The carrier phase measurement in units of meters (Φ) at the user u from satellite k on frequency m at epoch i is modeled as Φ k u,m(t i ) = λ m φ k u,m(t i ) = ru(t k i ) + Tu k (t i ) q1mi 2 u(t k i ) + λ m Nu,m k + δru(t k i ) + λ m b φu,m +c(δτ u (t i ) δτ k (t i )) + λ m b k φ m + ε k φ u,m (t i ), (2.4) where λ m is the wavelength of the carrier; ru, k is the user-satellite range; Tu k is the tropospheric delay; Iu k is the ionospheric delay depending on the ratio of frequencies q 1m ; δru k is the projected satellite orbital error; b φu,m is the receiver phase noise; b k φ m is the satellite phase bias; and ε k φ u,m is the phase noise including multipath. In contrast to the code measurements, the phase ionospheric delay is negative and the phase multipath is also different, which will be explained on the next section. Note that the multiplication of the phase noise and biases by the wavelength will be later stipulated on the notation b φu,m, b k φ m and ε k φ u,m. The carrier phase measurements are extremely precise, but affected by integer ambiguities, whose estimation can be performed by a variety of algorithms (e.g. LAMBDA, Three-Carrier Ambiguity Resolution (TCAR) (Forssell et al. [9]), Rounding,... etc.).

13 Chapter 2. Global Navigation Satellite Systems Measurement s errors and delays Receiver noise, multipath, errors in the navigation message from the satellite and atmospheric delays are different kind of error sources, which affect the precision of the GNSS measurements. Ephemeris and clock errors The satellite computed orbit or ephemeris is uploaded to the satellite to be broadcast in their navigation message over several days. It offers the states of the satellites and their clocks. Errors on the estimation and prediction of the ephemeris parameter values result in a false satellite predicted orbit. Therefore, the more accurate are the prediction models, and the more frequent the ephemeris uploads, the lower the sizes of the range errors. Atmospheric delays The earth s atmosphere modifies the speed and direction of propagation of the GNSS signals. This effect, referred to as refraction, generates a propagation delay, i.e. the signal transit time is changed. Two layers of the atmosphere affect particularly the propagation of the GNSS signals: the ionosphere and the troposphere. Ionospheric delay The ionosphere is the layer of the earth s atmosphere that is ionized by solar radiation. It extends from 50km to 1000km above the surface of the Earth. The UV radiation of the sun heats the neutral gas molecules, which then break and liberate free electrons and ions. This process is referred to as ionization. Depending upon the solar activity, the free electron density varies considerable. The ionosphere is a dispersive medium for radio signals, which means that the ionospheric delay depends on the frequency of the signal. The ionosphere effects are inversely proportional to the square of the frequency of the signal. The first order ionospheric delays can be estimated by collecting measurements on different frequencies q 2 1m = f 2 1 f 2 m = λ2 m. λ 2 1 The number of free electrons in the path of a signal affects the code and carrier phase measurements differently, but with the same magnitude. That is, in the ionosphere, the

14 Chapter 2. Global Navigation Satellite Systems 10 phase velocity of the GNSS carrier signals exceeds that of light in vacuum, whereas the group velocity is delayed. Therefore, the ionospheric phase delay is negative (see equation (2.4)), i.e. the phase is advanced. Tropospheric delay The troposphere, composed of dry gases and water vapor, is the lower layer of the earth s atmosphere, and generates also a signal delay. It extends from the surface of the Earth to about 9 km above the poles and about 16 km above the equator. In contrast to the ionosphere, the troposphere is a non-dispersive medium for GNSS frequencies, i.e. the propagation delay does not depend on the frequency of the signal. Furthermore, the phase and group velocities are the same; therefore the tropospheric delays are equal for the code and carrier phase measurements. The tropospheric delay can not be estimated from GNSS measurements, consequently tropospheric models and mapping functions are used to correct for it. Typically, the tropospheric delay is separated into a wet delay T w, caused by the water vapor; and a dry delay T d, caused by dry air and some water vapor, so that T = T d + T w. The tropospheric wet delay is much harder to model than the dry delay, which is fortunately larger and can be predicted with higher accuracy. Random measurement noise The code and carrier phase measurements are affected also by random errors unrelated to the signal, e.g. noise introduced by the antenna, signal quantization noise and interference from other signals. This noise is also called receiver noise and varies with the signal strength. Multipath Multipath refers to the effect of a signal arriving at an antenna via two or more paths due to the reflections of the direct path (i.e. line-of-sight) from buildings, structures and from the ground. The reflected signals are delayed and usually weaker than the direct signal. The magnitudes of the errors induced by multipath on the code and carrier phase measurements differ significantly. In the code measurements, it varies between 1 m and 5 m, whereas the

15 Chapter 2. Global Navigation Satellite Systems 11 corresponding error in the carrier phase measurements is 1 5 cm. Moreover, the multipath error in the carrier phase measurements does not exceed a quarter cycle if the amplitude of the reflected signal is smaller than the amplitude of the direct signal.

16 Chapter 3 Carrier Phase Positioning and Integer Ambiguity Resolution The precision and accuracy given by the code and carrier phase measurements are very different. The carrier phase measurements offer a very-high precision (order of millimeters), whereas the code measurement s precision is in the order of meters. Carrier phase measurements are therefore required for precise positioning. However, they are ambiguous (integer ambiguities); consequently, integer ambiguity resolution is necessary before the measurements can be used for precise positioning. Integer ambiguity resolution consists of an estimation and a validation part [10]. This means that given the estimates of the integers, it is necessary to prove their precision and accuracy in order to obtain precise position estimates. In the following section, some methods to improve the performance of ambiguity resolution for absolute carrier phase positioning are described. Moreover, an overview of some usually ambiguity resolution methods will be given on Section 3.3. Finally, a form to calculate their probability of correct estimation is shown on Section Simplification methods for ambiguity resolution Usually, in the case of precise relative positioning, ambiguity resolution is done by using double-difference (DD) carrier phase measurements (see Appendix A), which are used to reduce nuisance parameters on the measurements. However, another methods, discussed below, can also be used to improve the performance of ambiguity resolution, and for absolute carrier phase positioning as will be discussed on Chapter 4. 12

17 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution Linear combinations Integer ambiguity resolution of the carrier phase measurements can be simplified by using linear combinations of the measurements at multiple frequencies. The design of ionospherefree mixed code-carrier linear combinations of minimum noise and maximum combination discrimination for Galileo has been performed by Henkel and Günther in [11] and [12]. An overview of the design and characteristics of these combinations is described below. Furthermore, this work has been extended to four-frequency Galileo combinations (E1-E5a- E5b-E6), which design and comparison to other combinations is introduced on Section Code-carrier linear combinations Given the carrier phase measurements from Equation (2.4) and the code measurements from Equation (2.3), a linear combination of them at multiple frequencies can be done in order to obtain a larger wavelength and a low noise level, while the ionospheric delay is eliminated. Hence, the carrier phase and code measurements should be weighted by the corresponding coefficients, α m and β m. An illustration of this is shown in Figure 3.1. The properties of the linear combination at M frequencies are regulated by doing some constraints on the weighting coefficients: Geometry-preserving (GP): The user-satellite range r should be preserved by M α m + m=1 M m=1 β m! = 1, (3.1) where the phase part M m=1 α m will be weighted by the carrier phase geometric weight τ lc, and the code part M m=1 β m by (1 τ lc ). This constraint refers also to the non-dispersive errors, i.e. the satellite orbital errors, the satellite clock offsets and the tropospheric delays are not amplified. Ionosphere-free (IF): The ionospheric delay I of first order is eliminated if M α m q1m 2 m=1 M β m q1m 2! = 0. (3.2) m=1 Integer-preserving (NP): The combination of N m ambiguities should be an integer multiple of a common wavelength λ lc, that is M m=1 α m λ m N m! = λ lc N, (3.3)

18 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 14 Figure 3.1: Code-carrier linear combination scheme. which can be split into the next integer conditions where Z denote the space of integers. j m = α mλ m λ lc Z, (3.4) The wavelength of the code-carrier combination is obtained as follows λ lc (τ lc ) = τ lc M m=1 j m = λτ 1 lc with λ = M j m. (3.5) λ m m=1 λ m A combination discrimination indicator is given as a cost function in order to select the linear combinations. This has been defined by Henkel and Günther in [11] as D = λ lc 2σ n, (3.6) with the overall noise contribution of the linear combination σ n = M M αmσ 2 φ 2 m + βmσ 2 ρ 2 m, (3.7) m=1 where σ ρm are the code noises; and σ φm is the phase noise, which can be assumed to be equal for the Galileo frequencies (E1, E5b, E5b, E6) due to their close vicinity. m=1 The code noises σ ρm are obtained from the Cramér Rao Bound (CRB) given by Γ m = C N 0 c 2 (2πf) 2 S m(f) 2 df Sm(f) 2 df, (3.8) where c is the speed of light; C/N 0 is the carrier to noise power ratio; and S m (f) is the power spectral density, which has been derived for binary offset carrier (BOC) modulated signals

19 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 15 Table 3.1: Cramér Rao Bounds for Galileo signals. Frequency band Signal Bandwidth [MHz] CRB [cm] E1 CBOC(6,1,1/11) E5 (E5a+E5b) AltBOC(15,10) E5a BPSK(10) E5b BPSK(10) E6 BOC(10,5) by Betz in [13]. Table 3.1 shows the Cramér Rao Bounds at a carrier to noise power ratio C/N 0 = 45dB/Hz for the wideband Galileo signals. Two kinds of optimization criteria for this indicator can be done by using the remaining degrees of freedom of the linear combination (e.g. τ lc, code weighting coefficients β m ), after fulfilling the constraints mentioned above. One optimization criterion is to maximize the combination discrimination D, which is then used to select the linear combination that minimizes the probability of wrong fixing over all linear combinations. The probability of wrong fixing computation will be discussed on Section 3.4. On the other hand, the other criterion is to minimize the noise variance σ 2 n. For both criteria, a numerical search over the integers j m should be also included. For simplicity, the time and user indices are omitted and the index lc is used instead; and the resulting code-carrier linear combination is written as Φ k lc = λ lc φ k lc = r k + T k + λ lc N k lc + δr k +c(δτ δτ k ) + b φlc + b k φ lc + ε k φ lc, (3.9) where λ lc is the wavelength of the linear combination; and b φlc = M m=1 α mb φm + M m=1 β mb ρm. Note that the ionospheric delay is eliminated, and the obtained large wavelength λ lc increases the reliability of ambiguity resolution. Code-only linear combinations The reliability of ambiguity resolution is further improved by an additional linear combination of the code measurements from Equation (2.3), referred as to code-only combination, which does not add further ambiguities. Figure 3.2 depicts the linear combination. The properties of the linear combination at M frequencies are regulated by a geometry-preserving and ionosphere-free constraints, as for the code-carrier linear combinations, but without the

20 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 16 Figure 3.2: Code-only linear combination scheme. carrier phase weighting coefficients. The corresponding constraints on the code weighting coefficients are then written as Geometry-preserving (GP): Ionosphere-free (IF): M m=1 m=1 b m! = 1, (3.10) M b m q1m 2! = 0. (3.11) The code weighting coefficients b m are represented in Roman letters in order to differentiate them from the corresponding code weighting coefficients of the code-carrier linear combination. The noise variance of the code-only linear combination, which should be minimized as optimization criterion, is given by σ 2 n = M b 2 mσρ 2 m, (3.12) m=1 with the code noises σ ρm, that can be also obtained as explained in the code-carrier linear combinations. For simplicity, the time and user indices are omitted and the index lc is used instead; and the resulting code-only linear combination is written as ρ k lc = r k lc + T k lc + δr k lc + c(δ lc τ δ lc τ k ) + b ρlc + b k ρ lc + ε k ρ lc, (3.13) where the ionospheric delay is also eliminated.

21 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 17 Carrier phase-only linear combinations The design of the carrier phase-only combination follows the same constraints as for codeonly combinations, i.e. the corresponding equations can be obtained by replacing the code weighting coefficients b m of Equations (3.10) and (3.11) with the carrier phase weighting coefficients a m. The integer preserving (NP) constraint can be also applied for the resulting ambiguities, but not necessarily. The noise variance of the carrier phase-only linear combination, which should be minimized as optimization criterion, is given by M σn 2 = a 2 mσφ 2 m, (3.14) m=1 with the phase noises σ φm, that can be also obtained as explained in the code-carrier linear combinations. This carrier-phase only combination can be useful to reduce the noise of the other linear combinations, as will be introduced on Section GP-IF-NP Four-frequency Galileo code-carrier combination Given the code and carrier phase measurements from Equations (2.3) and (2.4), an ionosphere-free four-frequency Galileo (E1-E5a-E5b-E6) linear combination has been computed. Two different wavelengths with a noise level of a few centimeters were found: m and m. The analytic determination of this combination and comparison of its properties to other dual and triple frequency combinations is described below. Analytic determination The properties of the linear combination are regulated by applying the geometry-preserving (GP), ionosphere-free (IF) and integer-preserving (NP) constraints from Equations (3.1) to (3.4) with M = 4 on the weighting coefficients. In this case, the corresponding phase weighting coefficients can be rewritten to α 1 = j 1λ lc λ 1, α 2 = j 2λ lc λ 2, α 3 = j 3λ lc λ 3, α 4 = j 4λ lc λ 4, (3.15) where λ 1 is the wavelength of the frequency E1; λ 2 is the wavelength of the frequency E5a; λ 3 is the wavelength of the frequency E5b;and λ 4 is the wavelength of the frequency E6.

22 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 18 The wavelength λ lc of the combination will be obtained by replacing these coefficients in Equation (3.5). That is λ lc = where τ lc = α 1 + α 2 + α 3 + α 4. τ lc j 1 λ 1 + j 2 λ 2 + j 3 λ 3 + j = λτ 1 with λ 4 = j 1 λ 4 λ 1 + j 2 λ 2 + j 3 λ 3 + j 4 λ 4, (3.16) For this four frequency linear combination there exist 5 degrees of freedom: 4 for the code weights β m and 1 for the wavelength λ lc. The first two of them, β 1 and β 2, were required to fulfill the geometry-preserving (3.1) and ionosphere-free (3.2) constraints, and can be analytical derived as with β 1 = 1 w 2 (w 1 + 1)τ lc (w 3 + 1)β 3 (w 4 + 1)β 4, (3.17) β 2 = w 1 τ lc + w 3 β 3 + w 4 β 4 + w 2, (3.18) w 1 = k+1 q12 2 1, w 2 = 1 q12 2 1, w 3 = 1 q2 13 q12 2 1, w 4 = 1 q2 14 and k = λ 4 j m q m=1 λ m q1m, 2 where q 1m is the corresponding ratio of frequencies m = E5a, E5b and E6 with respect to the frequency E1. The remaining degrees of freedom β 3, β 4 and τ lc will be used to maximize the combination discrimination (see Equation 3.6), which is rewritten into D = λ lc 2σ n = λ lc, (3.19) 2 σ 2φ (α2 1 + α2 2 + α3 2 + α4) 2 + β1σ 2 2ρ1 + β2σ 2 2ρ2 + β3σ 2 2ρ3 + β4σ 2 2ρ4 where the phase noise σ φ will be assumed to 1 mm and 2 mm; and the code noises σ ρm obtained from the Cramér Rao Bounds (CRB) at a carrier to noise power ratio C/N 0 = 45dB/Hz from Table 3.1. The optimization criterion includes a numerical search over the integers j m which has been limited to j m 5 as all the other combinations suffer from biases and noise amplification. Moreover, the following optimization constraints are applied to the remaining degrees of freedom D 2 (β 3, β 4, τ lc ) τ lc! = 0, D 2 (β 3, β 4, τ lc ) β 3! = 0 and are D 2 (β 3, β 4, τ lc ) β 4! = 0. (3.20) The results of the analytical derivation of the code weight β 3 and the carrier phase geometric weight τ lc from Equation (3.20) are described as follows τ lc = (2c 1β c 2 β c 5 β 3 β 4 + 2c 7 β 3 + 2c 8 β 4 + 2c 9 ) c 3 β 3 + c 4 β 4 + c 6, (3.21)

23 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 19 where c 1 = σρ 2 1 (w 3 + 1) 2 + σρ 2 2 w3 2 + σρ 2 3, c 2 = σρ 2 1 (w 4 + 1) 2 + σρ 2 2 w4 2 + σρ 2 4, c 3 = 2(σρ 2 1 (w 1 w 3 + w 1 + w 3 + 1) + σρ 2 2 w 1 w 3, c 4 = 2(σρ 2 1 (w 1 w 4 + w 1 + w 4 + 1) + σρ 2 2 w 1 w 4, c 5 = 2(σρ 2 1 (w 3 w 4 + w 3 + w 4 + 1) + σρ 2 2 w 3 w 4, c 6 = 2(σρ 2 1 (w 1 w 2 w 1 + w 2 1) + σρ 2 2 w 1 w 2, c 7 = 2(σρ 2 1 (w 2 w 3 + w 2 w 3 1) + σρ 2 2 w 2 w 3, c 8 = 2(σρ 2 1 (w 2 w 4 + w 2 w 4 1) + σρ 2 2 w 2 w 4, c 9 = σρ 2 1 (1 w 2 ) 2 + σρ 2 2 w2; 2 with the coefficients β 3 = s 1β 2 4 s 2 β 4 + s 3 s 4 β 4 + s 5, (3.22) s 1 = c 4 c 5 2c 2 c 3, s 2 = 2c 3 c 8 c 5 c 6 c 4 c 7, s 3 = c 6 c 7 2c 3 c 9, s 4 = c 3 c 5 2c 1 c 4, s 5 = c 3 c 7 2c 1 c 6. The code weight β 4 is also analytically derived from Equation (3.20) as v 1 β4 3 + v 2 β4 2 + v 3 β 4 + v 4 = 0, (3.23) with the coefficients v 1 = s 2 1s 5 + r 3 s s 1 s 2 s 4 + r 1 s 1 s 4, v 2 = 2r 3 s 4 s 5 + r 1 s 1 s 5 s 1 s 3 s 4 s 1 s 2 s 5 r 1 s 2 s 4 s 2 2s 4 r 2 s 2 4, v 3 = r 3 s s 1 s 3 s 5 + r 1 s 3 s 4 + 2s 2 s 3 s 4 2r 2 s 4 s 5 r 1 s 2 s 5, v 4 = s 2 3s 4 + r 2 s 2 5 r 1 s 3 s 5, where r 1 = 2c 4 c 7 c 5 c 6 c 3 c 8, r 2 = c 6 c 8 2c 4 c 9, r 3 = c 4 c 8 2c 2 c 6. Table 3.2: Four-frequency code-carrier linear combinations (σ φ = 1 mm, σ ρm = Γ m ). E1 E5a E5b E6 λ σ n D α m j m βm α m j m βm m 6.34cm cm 2.48mm 21.9 The non-linear optimization can be obtained after solving for the non-linear equation of the code weight β 4, which can be solved analytically by using the Cardano s method (see

24 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 20 Appendix B). The resulting weighting coefficients and properties of the four frequency codecarrier linear combinations of maximum discrimination with a phase noise σ φ = 1 mm are shown in Table 3.2. The first combination refers to a widelane, while the second one to a narrowlane combination. The higher E1 code noise is significantly suppressed, which benefits by a larger wavelength of m and a lower noise level of 6.34 cm. Table 3.3: Four-frequency code-carrier linear combinations (σ φ = 2 mm, σ ρm = 3 Γ m ). E1 E5a E5b E6 λ σ n D α m j m βm α m j m βm m 13.7cm cm 4.97mm 10.9 Table 3.3 shows the resulting weighting coefficients and properties of the four-frequency code-carrier linear combinations of maximum discrimination with a phase noise σ φ = 2 mm and a Cramér Rao Bound scaled by a factor of three to include fast varying multipath. A different widelane combination is found with a large wavelength of m and a noise level of 13.7 cm. Table 3.4: GP-IF-NP code-carrier widelane combinations of maximum discrimination for two, three and four frequencies (σ φ = 2 mm, σ ρm = 3 Γ m ). E1 E5 E5a E5b E6 λ σ n D α m j m βm α m j m βm α m j m βm α m j m βm m 19.0cm m 34.0cm m 11.9cm m 13.7cm 15.6 The trade-off between the wavelength and the noise level for a limited numerical search over the integers j m for four-frequency (E1-E5a-E5b-E6) widelane and narrowlane linear combinations is illustrated on Figure 3.3. The respective properties of the combinations that maximize the combination discrimination indicator are showed on Table 3.5.

25 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 21 The additional use of E6 measurements offers the advantage of using its lower code noise as the main noise contribution to the linear combination, which benefits by a larger wavelength an a lower noise level. This increases the combination discrimination by more than 50% in comparison to another optimized linear combinations with two and three Galileo frequencies. Table 3.4 depicts the weighting coefficients and properties of code-carrier widelane linear combinations of maximum discrimination for two, three and four Galileo frequencies (Henkel et al. [14]). 1 Noise standard deviation σ n [m] j m 3 j m 2 j m Wavelength λ [m] (a) Widelane Noise standard deviation σ n [m] j m 2 j m Wavelength λ [m] (b) Narrowlane Figure 3.3: Four-frequency code-carrier linear combinations for a limited numerical search over the integers j m. (σ φ = 2 mm, σ ρm = 3 Γ m )

26 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 22 Table 3.5: GP-IF-NP four-frequency code-carrier widelane and narrowlane combinations of maximum discrimination for a limited numerical search over the integers j m. (σ φ = 2 mm, σ ρm = 3 Γ m ) E1 E5a E5b E6 λ σ n D α m j m βm α m j m βm α m j m βm α m j m βm m 14.1cm m 22.1cm cm 12.67mm cm 23.03mm Carrier smoothing The noise and multipath of the code-carrier and code-only combinations can be reduced by smoothing them with a carrier phase-only combination of minimum noise variance. A non-recursive Hatch filter [15] variant introduced by Hwang et al. [16], which is illustrated on Figure 3.4, will be used to obtain the smoothed linear combination Ψ sm. An ionospherefree code-carrier combination of arbitrary wavelength or a code-only combination can be used as the noisy upper input Ψ A, while an ionosphere-free carrier-phase-only combination is used as lower input Φ B. Figure 3.4: Carrier smoothing process. The smoothed linear combination is written as Ψ sm (t) = χ(t) + Φ B (t), (3.24)

27 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 23 with χ(t) = Ψ A (t) Φ B (t), which is filtered by a low pass filter, i.e. ( χ(t) = 1 1 ) χ(t 1) + 1 χ(t). (3.25) τ τ The variance of the smoothed linear combination which can be derived by using the low pass filter equations and the geometric series n=0 qn = 1/1 q, is given by Henkel and Günther in [11] as σsm 2 = σb τ s 1 (σ2 A 2σ AB + σb) (σ AB σ τ B), 2 (3.26) s where τ s is the smoothing time constant; σb 2 is the variance of the carrier phase-only combination from Equation (3.14); σa 2 is the variance of the combination to be smoothed (e.g. code-carrier); and σ AB is the covariance between both combinations. Note that the use of carrier smoothing reduces the noise variance, which limits the margins for the biases, but having no impact on them; however, the main cost is the time needed for the desired accuracy. Also, the ambiguities of the carrier phase-only combination have no impact on the smoothed output as they are cancelled by the use of different signs in the addition Satellite-satellite single-difference (SD) In order to eliminate the receiver biases and clock errors satellite-satellite single-difference (SD) measurements can be utilized. This simplifies the measurements and also improves their precision. For simplicity, the time indices are omitted, and the code and carrier phase satellite-satellite (kl) SD measurements obtained from Equations (2.3) and (2.4) are modeled as ρ kl u,m = ρ k u,m ρ l u,m = ru kl + Tu kl + q1mi 2 u kl + b kl ρ u,m + ε kl ρ u,m, (3.27) Φ kl u,m = Φ k u,m Φ l u,m = ru kl + Tu kl q1mi 2 u kl + λ m Nu,m kl + λ m b kl φ u,m + ε kl φ u,m, (3.28) where the corresponding SD clock offsets c(δτ kl ) and the SD projected orbital errors δr kl u have been mapped respectively to the SD code biases b kl ρ u,m and SD phase biases b kl φ u,m. 3.2 Linear model for position estimation The user-satellite range r k u obtained from the GNSS measurements can be expressed as a function of the satellite and the user positions as r k u = x k x u, (3.29)

28 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 24 where x k = (x k, y k, z k ) T is the satellite position vector; x u = (x u, y u, z u ) T is the user position vector; and denotes the magnitude of a vector. Given the code and carrier phase measurements, a linearization of them with respect to the user position can be done by using approximate values of all the parameters, that means a computed measurement is generated. For simplicity, it will be assumed that all the approximate values except for the user-satellite range are zero. Let x k 0 = (x k 0, y0, k z0) k T be the approximate satellite position and x 0 = (x 0, y 0, z 0 ) T the approximate user position, then the corresponding range approximation is r0 k = x k 0 x 0, (3.30) and the resulting observed-minus-computed equations are given by δφ k u,m = Φ k u,m r0 k and δρ k u,m = ρ k u,m r0. k (3.31) Considering that the true position of the user can be represented as x u = x 0 + δx, with δx being a correction for the estimate, then the difference in the range can be linearized as follows δr k = r k u r k 0 = x k 0 x u x k 0 x 0 = x k 0 x 0 + δx x k 0 x 0 xk 0 x 0 x k 0 x 0 δx = 1k δx, (3.32) where 1 k is the unit vector pointing from the user to the satellite and a Taylor series approximation have been used. By substituting the linearized range on the observed-minus-computed equations gives the following linearized equations y ρ = δρ k u,m = 1 k δx + Tu k + q1mi 2 u k + δru k +c(δτ u δτ k ) + b ρu,m + b k ρ m + ε k ρ u,m, (3.33) y φ = δφ k u,m = 1 k δx + Tu k q1mi 2 u k + λ m Nu,m k + δru k +c(δτ u δτ k ) + b φu,m + b k φ m + ε k φ u,m. (3.34) A GNSS mathematical model, also known as the Gauss-Markov model, can be obtained from the last linearized equations and is given in a general form as E{y} = Ax; D{y} = E{εε T } = Σ y, (3.35)

29 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 25 where E{ } is the expectation operator; D{ } is the dispersion operator; y is the vector of measurements; A is the design matrix; x is the vector of parameters; ε is the discrepancy between y and Ax; and Σ y is the noise covariance matrix of the measurements. The first part of the model (functional model) describes the relation between the measurements and the parameters, while the second part (stochastic model) describes the noise characteristics of the measurements; therefore Σ y is needed as weight of the measurements for the least-squares estimation of the unknown parameters, which will be discussed in the following section. 3.3 Integer estimation The GNSS mathematical model from equation (3.35) can be parameterized in integers and real-values as y = A 1 N + A 2 u + ε, N Z n, u, ε R n (3.36) where the vector N consists of the unknown integer ambiguities; the vector u consists of the remaining unknown parameters (e.g. user position, tropospheric delay); and ε is the noise vector. A commonly used method to solve this model is the least-squares criterion, which looks for the estimates which minimizes the cost function C(N, u) = y A1 N A 2 u 2. (3.37) The cost function is the sum of the lengths of residuals squared, which would be directly to calculate if the integer constraint on each element of N were not considered. Hence, an integer-least squares problem arises, which can be solved by giving different weights to the residuals by using the inverse of the noise covariance matrix of the measurements Σ 1 y. It reads C Σ 1 y (N, u) = y A1 N A 2 u 2 Σ 1 y = (y A 1 N A 2 u) T Σ 1 y (y A 1 N A 2 u). (3.38) The procedure to solve this minimization problem can be divided into three steps (Teunissen [1]) as follows: 1. Float solution: The integer constraint of the ambiguities N is disregarded (i.e. N R n ), and a float solution (denoted by a hat sign ˆ ) which minimizes the equation (3.38) is obtained.

30 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution 26 The resulting real-valued estimates and their covariance matrix are written as ( ) ( ) ˆN Σ ; Σ float = ˆN Σ ˆN,û. (3.39) û Σû Σû, ˆN 2. Ambiguity resolution: Given the float ambiguity vector ˆN from Equation (3.39) compute the integer ambiguity vector N which minimizes the cost function C(N) = ˆN N 2 Σ 1 ˆN = ( ˆN N) T Σ 1 ˆN ( ˆN N). (3.40) A mapping S from the space of reals to the space of integers is needed to obtain the integer solution (denoted by a check sign ˇ ) to Equation (3.40), this means Ň = S( ˆN), where S : R n Z n. (3.41) Since the space of integers Z n is discrete, the mapping will be a many-to-one map, this means that many real-valued ambiguity vectors N will be mapped to the same integer vector. Therefore, a subset S z R n, also known as pull-in region by Teunissen [17] and Jonkman [18], can be assigned to each integer vector z Z n as follows S z = {x R n z = S(x)}, z Z n. (3.42) Then, the integer estimator can be expressed as Ň = z Z n zs z ( ˆN), with s z (x) = { 1 if x Sz 0 otherwise (3.43) where s z (x) is an indicator function. Different integer estimators can be chosen for the mapping S. A brief overview of them will be given on the following section. 3. Fixed solution: The integer ambiguity estimates are used to correct the float estimates of the remaining parameters û, it reads û(ň) = û Σ û, ˆN Σ 1 ˆN ( ˆN Ň) = ǔ. (3.44)

31 Chapter 3. Carrier Phase Positioning and Integer Ambiguity Resolution Integer estimators The float ambiguity solution ˆN from Equation (3.39) can be represented as a vector of estimates with the corresponding covariance matrix as ˆN 1 σ 2ˆN1 σ 2ˆN1... σ 2ˆN1 ˆN2 ˆNn ˆN 2 ˆN =, Σ. ˆN = σ 2ˆN2 σ 2ˆN2... σ 2ˆN2 ˆN1 ˆNn.. (3.45)..... ˆN n σ 2ˆNn σ 2ˆNn... σ 2ˆNn ˆN1 ˆN2 This notation of the float ambiguity solution will be used to describe the following integer estimators. Integer Rounding Integer rounding is the simplest integer estimator. The integer solution is obtained by rounding each of the entries of the float solution ˆN to their nearest integer. The corresponding integer estimation procedure is written then as [ ˆN 1 ] [ ˆN 2 ] Ň R =, (3.46). [ ˆN n ] where [ ] denotes rounding to the nearest integer. This estimator does not take the ambiguity correlation into account; therefore, the solution will not always satisfy the cost function formulated in Equation (3.40), unless the ambiguity covariance matrix Σ ˆN were a diagonal matrix (i.e. there is no correlation between the ambiguities). Integer Bootstrapping In contrast to integer rounding, the integer bootstrapping (Blewitt [19]) estimator takes some of the correlation between the ambiguities into account. It is also knows as sequential integer rounding, because the integer solution is computed as follows: the first ambiguity ˆN 1 is rounded to its nearest integer. After this, the estimates of the remaining ambiguities are corrected by virtue of their correlation with the first ambiguity. Then the second (corrected)